---
tag: J72DEZHNAW
kind: conjecture
name: Inscribed square problem
msc: [51M04, 52A10]
uses: [045YFKRW0P]
about: [045YFKRW0P]
sources:
  - url: https://en.wikipedia.org/wiki/Inscribed_square_problem
same_as:
  wikidata: Q2264221
  wikipedia: Inscribed square problem
history:
  posed_by: "Otto Toeplitz"
  first_stated: "1911"
---

## Statement

Let $C \subset \R^2$ be a Jordan curve, that is, a [[closed curve|045YFKRW0P]] in the plane that does not meet itself.
Then $C$ contains four points that are the vertices of a square.

## Remarks

The conjecture is open in general.
It is known for many classes of curves, among them convex curves, polygons, and curves that are smooth enough.
For smooth Jordan curves, Joshua Greene and Andrew Lobb proved more: such a curve contains the vertices of a rectangle of every prescribed aspect ratio.
The difficulty for general continuous curves is that limits of inscribed squares of approximating smooth curves may shrink to a single point.

## Explanations

### L1 human

Draw any closed loop on paper that never crosses itself, however wiggly.
The question is whether one can always find four points on the loop that form the corners of a square.
For a circle this is easy, and for many nice curves it has been proved, but for arbitrary continuous loops nobody knows.